ARTICLE

https://doi.org/10.1057/s41599-021-00806-w OPEN A balance for fairness: fair distribution utilising physics ✉ Song-Ju Kim1,2 , Taiki Takahashi3 & Kazuo Sano1,4

The problem of ‘uneven distribution of wealth’ accelerated during the COVID-19 pandemic. In the chaotic modern society, there is an increasing demand for the realisation of true ‘fairness’. In this study, we propose a fair distribution method ‘using physics’, which imitates the Greek ‘ ’ fi 1234567890():,; mythology, Themis, having a balance of judgement in her left hand, for the pro t in games of characteristic function form. Specifically, we show that the linear programming problem for calculating ‘nucleolus (a solution for the fair distribution)’ can be efficiently solved by con- sidering it as a physical system in which gravity works. In addition to significantly reducing the computational complexity, the proposed scheme provides a new perspective to open up a physics-based policymaker that is adaptable to real-time changes. We will be able to implement it not only in liquid systems but also in many other physical systems, including semiconductor chips. The fair distribution problem can be solved immediately using physical systems, which should reduce disputes and conflicts based on inaccurate information and misunderstandings, eliminating fraud and injustice.

1 SOBIN Institute LLC, Kawanishi, Japan. 2 Graduate School of Media and Governance, Keio University, Fujisawa, Japan. 3 Department of Behavioral Science, Research and Education Center for Brain Sciences, Center for Experimental Research in Social Sciences, Hokkaido University, Sapporo, Japan. 4 Department of ✉ Economics, Fukui Prefectural University, Fukui, Japan. email: [email protected]

HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w 1 ARTICLE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w

Introduction he Buddha taught that the ‘goodness’ that embodies Previously, Kim proposed a flexible analogue computation ‘righteousness’ stands in contrast to duties and to personal scheme that overcomes the computational difficulties of digital T fi ‘ ’ ties. In other words, the de nition of good is what will be computing by taking advantage of natural (physical) properties, in the interest of oneself, in the interest of others and in the such as conservation laws, continuity and fluctuations (Kim et al. interest of those to be born in the future. Humans feel happy 2016, 2017). In particular, he extracted an efficient decision- when they are needed within a community and when they play a making scheme (reinforcement learning) called the ‘tug-of-war role that benefits others and the whole community. This core principle’ (Kim et al. 2015), and he and coworkers created ‘self- aspect of human nature is often forgotten in actual social activ- decision-making physical systems’ by applying it to quantum dots ities, seen as a ‘beautiful thing’ at a distance from the practical. (Kim et al. 2013), single photons (Naruse et al. 2015), atomic In a modern society driven by neoliberalism, only the more switches (Kim et al. 2016), semiconductor lasers (Naruse et al. ‘primitive’ desires of man are judged essential. Because neoliberalism 2017), ionic devices (Tsuchiya et al. 2018), resistive memories assumes that individual negligence is the source of poverty, disparity (Kim et al. 2019) and internet of things (IoTs) (Ma et al. 2019). and poverty are the incentives for hard work rather than goodness or The physics-based computations are useful in the calculation of spiritual insight (Jinno 2010). We have come to believe that market- ‘fairness’. Themis, the Greek goddess of justice, installed in based competition brings efficiency and optimisation to society. As a courthouses and law offices, is considered the ‘guardian of justice result of this, not only the destruction of the natural environment and order’ (see Fig. 1). She wears a blindfold as a sign of her but also the destruction of the human environment has been pro- objectivity and selflessness, holding a sword in her right hand to moted as a means of achieving growth. The result has been a loss of protect society from vice and a ‘balance of justice’ in her left hand trust and bonds between humans due to excessive competition. to measure right and wrong. In other words, she leaves the Today, humanity is facing a serious crisis not only because of impartial judgment to nature (physical phenomena) rather than the COVID-19 but also because of the lack of trust between perceptions and attachments. Although that balance is a symbol, people. In order to defeat infectious diseases, people need to trust the ‘use of nature (physics)’ is thought to have the effect of scientists, citizens need to trust public institutions and countries eliminating artificiality and gaining consensus (concessions) from need to trust each other. New viruses can occur everywhere, and the people concerning issues. if they do occur, the important question will be whether the The direct use of physics in computation is not a particularly government can immediately implement a city blockade. new concept (Steinhaus 1960; Levi 2009). Rather, before the Unfortunately, we must include the economic loss of the blockade advent of the digital computer, everything was analogue. For as a variable. In such a case, the timing of the blockade may be example, suppose that Fig. 2 has three villages with 50, 70 and 90 delayed without international assistance (Harari 2020). elementary school students living in each village. How can we Nevertheless, it has been proven in socialist countries that easy minimise the total distance travelled for all students’ to school solutions to poverty, such as equal distribution, do weaken when we build an elementary school? As shown in Fig. 2, the incentives for hard work. Also, the fact that acting alone and centre of gravity is the answer (as to where the elementary school selfishly within an altruistic community can produce great ben- should be built) made by hanging weights determined by the efits for the individual (in the short term) seems to be an obstacle number of students in each village. If we consider the brute-force to formulating a powerful critique of neoliberal ideology. The search for possible locations, the calculation would be intractable, question that remains is whether it is not possible to build a however, by ‘designing’ the problem in this way, we can leave the system that mobilises digital and new analogue technologies so calculation (or even part of it) to physics. that people can do ‘good’ with confidence? The role of computation in our society is increasing daily. The problem of ‘uneven distribution of wealth’ (Piketty 2014) Humanities and social sciences are no exceptions, especially as has been accelerated in the midst of the COVID-19 pandemic computation, including artificial intelligence (AI), is transforming (Pastreich 2019, 2020). The ‘universal basic income’, which our society in many ways. For example, studies on computational includes wealth redistribution, solving social security problems policymaking, using AI and multi-agent simulations, have already and combating poverty, is being actively discussed and experi- begun (Lempert 2002; Zheng et al. 2020). Attempts to create a fair mented with. In the year 2020 Spain became the first country in voting system are being promoted (Brandt et al. 2016; Endriss the world to implement a ‘basic income’ for the needy. In this 2017). That includes voting theory, such as the computational chaotic ‘new turn of capitalism’, there is an urgent need to complexity of the winner determination and manipulation in establish a firm and blameless ‘fairness’ in order to avoid elections, algorithms for fair allocation and coalition formation, inequities and social justice for everyone. However, there are such as matching and hedonic games. Moreover, studies are also different kinds of difficulties for the realisation as follows. underway on building reciprocal relationships that people need to move from competition to co-operation (Nowak et al. 2006; 1. Fairness requires proper judgement and the processing of Yamamoto et al. 2020). various factors impacting the current situation. The Despite the importance of computation, there are serious calculation often cannot keep up with the changing limitations to the existing computational methods, including AI. dynamic system. When it comes to our ability to computing, we are almost 2. The elements critical to determining fairness that depends fi powerless, especially when considering solving societal problems. on human interpretation are dif cult to formalise. On the basis of four simple examples, this study shows that the 3. As Arrow’s impossibility theorem (Arrow 1951;Saeki1980) ‘ ’ calculation of fair distribution can be realised directly by shows, we do not even know whether there is a fair solution exploiting physics. If it is implemented on a semiconductor chip or not. (using analogue resistance values (Kim et al. 2019), it could be In this study, we propose a physical method to realise a ‘fair linked to GPS and big data, so that the distance from your current distribution of profit’ within the restricted domain of games. location to your home is immediately known, helping us decide in Therefore, we only have to focus on how to overcome the an instant how much to spend on a taxi in case of riding with computational difficulty for ‘fairness’ calculations as long as we friends (taxi problem), enabling us to make our own decisions. treat the restricted domain of games called ‘games in character- The distribution of the bankruptcy problem can also be calculated istic function form’. immediately, which should reduce disputes and conflicts based on

2 HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w ARTICLE

Fig. 2 An example for physically solving computational problems. There are three villages with 50, 70 and 90 elementary school students living in each village. How can we minimise the total distance travelled for all students' to school when we build an elementary school? The answer is the centre of gravity (as to where the elementary school should be built). This figure was reprinted with the permission of the copyright holder, the Oxford University Press. H. Steinhaus: Mathematical Snapshots, Oxford Univ. Press (1960).

within the games in characteristic function form (Neumann and Morgenstern 2007; Gilles 2010). Two types of games called the ‘taxi problem’ and the ‘bankruptcy problem’ are introduced. A cooperative game wherein the players can cooperate with each other can be an effective means of determining how the profit obtained through cooperation can be fairly distributed (Yokoo et al. 2013; Kishimoto 2015). Cooperative games have been applied to market analysis, voting analysis and cost-sharing analysis in economics, as well as to market design, such as auction theory and matching theory, the theme of the 2020 Nobel Prize in Economics (Kishimoto 2015). Let us define N = {1, 2, …, n}asthesetofn players. We call the non-empty subset of N ‘coalitions’.IfR is a set of real numbers, we call the real value function v:2N → R on 2N (the set of the subset of N) ‘a characteristic function’. For each coalition S (⊆N), v(S) represents the profit that can be gained by the members of the coalition S by cooperating. More concretely, it is the difference between the cooperative case and the non- cooperative case that is critical. We call (N, v), which is the set of players N and a characteristic function v,a‘game of characteristic function form’.

Taxi problem. Consider a situation in which three persons (1, 2, 3) finish eating at a restaurant and go home by taxi. If they return home alone, the charges are Person 1: $20, Person 2: $21 and Person 3: $25, respectively. In addition, between each of the two houses, the following taxi fees are charged: $10 between 1 and 2, $14 between 1 and 3 and $20 between 2 and 3. In this case, the Fig. 1 Themis, the Greek goddess of justice, installed in courthouses and shortest route is (2–1–3), which costs $45. fi ‘ ’ law of ces, is considered the guardian of justice and order . She wears a In this example, the characteristic functions are the followings. blindfold as a sign of her objectivity and selflessness, holding a sword in her right hand to protect society from vice and a ‘balance of justice’ in her left vð1; 2; 3Þ¼ð20 þ 21 þ 25ÞÀ45 ¼ 21; ð1Þ hand to measure right and wrong. This photograph was reprinted with the permission of the copyright holder, the Nippon Catalogue Shopping Co. Ltd. vð1; 2Þ¼41 À 30 ¼ 11; ð2Þ http://www.catalog-shopping.co.jp. vð1; 3Þ¼45 À 34 ¼ 11; ð3Þ inaccurate information and misunderstandings, eliminating fraud and injustice. vð2; 3Þ¼46 À 41 ¼ 5; ð4Þ

vð1Þ¼vð2Þ¼vð3Þ¼0: ð5Þ Games in characteristic function forms. In this study, we pro- pose a physical method to realise a ‘fair distribution of profit’ where a characteristic function v(x, y) is the profitifx and y have

HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w 3 ARTICLE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w cooperated, that is the difference between the total amount of The above expressions can be transformed to the followings fares for going home alone and the fare for going home in a taxi from Eqs. (16), (17), (18) and the total rationality Eq. (19). by the shortest route. À3T ≤ ðx þ x þ x Þ¼21 ≤ 3T þ 36 ð25Þ The question is, ‘How should the profitof$21 be divided 1 2 3 ‘ ’ ’ fairly among the three? In general, this problem can be solved () T ≥ À 7andT ≥ À 5 ð26Þ using the concept of ‘nucleolus (Schmeidler 1969)’ as follows. First, in the games of characteristic function form (N, v), we The minimum value of T that satisfies all these conditions define the ‘complaint’ C(S, x) of the coalition S with respect to the is −5, which, when substituted, yields the following constraint allocation x as follows. condition, C S; x v S ∑ x ≤ x ≤ ; ð Þ¼ ð ÞÀ i ð Þ À5 1 11 ð27Þ i2S 6 fi S ‘ ≤ x ≤ ; That is the pro t of the coalition minus the total allocation À5 2 5 ð28Þ (distribution) of those who participated in the coalition S’. The ‘ ’ ≤ x ≤ ; nucleolus is the allocation that minimises the largest complaint À5 3 5 ð29Þ and is generally known to be uniquely determined (for simplicity, x x x we avoid an exact mathematical definition of ‘nucleolus’ here). 1 þ 2 þ 3 ¼ 21 ð30Þ In the case of the taxi problem, the complaint against each T = − x = x = x = 5 can be realised only with 1 11, 2 5 and 3 5. coalition is as follows That is, ‘nucleolus’ in the taxi problem is (11, 5, 5) and according C ; ; x v ; x x x x ; ’ $ ðf1 2g Þ¼ ð1 2ÞÀ 1 À 2 ¼ 11 À 1 À 2 ð7Þ to the nucleolus s criterion, Person 1 would receive 11 and Persons 2 and 3 would both receive the allocation of $5. In C ; ; x v ; x x x x ; n ðf1 3g Þ¼ ð1 3ÞÀ 1 À 3 ¼ 11 À 1 À 3 ð8Þ general, in games of characteristic function form with players, we must solve a linear programming problem to minimise the C ; ; x v ; x x x x ; ðf2 3g Þ¼ ð2 3ÞÀ 2 À 3 ¼ 5 À 2 À 3 ð9Þ maximum complaint and if there is more than one allocation that achieves the smallest maximum complaint, the second-largest C ; x v x x ; fi ðf1g Þ¼ ð1ÞÀ 1 ¼À 1 ð10Þ complaint in the allocation must be minimised. To nd ‘nucleolus’, it is necessary to solve the linear programming C ; x v x x ; ðf2g Þ¼ ð2ÞÀ 2 ¼À 2 ð11Þ problems at most n − 1 time. The computational complexity of linear programming is commonly known as at the most C ; x v x x : ðf3g Þ¼ ð3ÞÀ 3 ¼À 3 ð12Þ polynomial time (Yokoo et al. 2013; Karmarkar 1984). x + x + x = v = Using the total rationality 1 2 3 (1, 2, 3) 21, we rewrite the complaints of the two-person coalitions. Bankruptcy problem (estate distribution problem). In what can be considered a game of characteristic function form, there exists a C ; ; x x ; ðf1 2g Þ¼ 3 À 10 ð13Þ famous problem known since long ago. If a person goes bankrupt leaving a debt of D , D , … , Dn to each of the n creditors 1, 2, … , n, C ; ; x x ; 1 2 ðf1 3g Þ¼ 2 À 10 ð14Þ how should the bankrupt’sentireestate(M)be‘fairly’ distributed to each creditor? At first glance, a proportional division according to C ; ; x x : ðf2 3g Þ¼ 1 À 16 ð15Þ thesizeofthedebtseemstobe‘fair’. However, if we consider a case Since the nucleolus is the allocation that minimises the in which the value of the entire estate is smaller than the amount of maximum constraint, we suppose the above six complaints any debt we encounter a situation in which, because all creditors are Eqs. (10), (11), (12), (13), (14) and (15) are less than or equal to a similarly unable to fully recover their share, it is better to split the certain value T, then we minimise T. In other words, consider the debt into equal parts. following linear programming problem. In fact, this sort of bankruptcy problem is described in the Minimise(T): Babylonian Talmud, a Jewish scripture more than 2400 years ago M D T ≤ x ≤ T ; (Table 1). In Table 1, represents the total estate and À 1 þ 16 ð16Þ represents the creditors and the amount of their debts. If the total estate is 300, the division is proportional, but if it is 100, it is an ÀT ≤ x ≤ T þ ; ð Þ 2 10 17 equal division. Even though for 200, the criteria are not as clear at first glance, there exists a clear criterion of ‘nucleolus’ which was ÀT ≤ x ≤ T þ ; ð Þ 3 10 18 first formulated in 1969. It is very surprising that the Sages of the Talmud more than 2400 years ago had clear ideas and criteria for x þ x þ x ¼ ; ð Þ 1 2 3 21 19 the concepts of ‘justice’ and ‘fairness’ in this specific financial case. We had to wait until 1985 for a satisfactory explanation of x ≥ ; ð Þ 1 0 20 this criterion of bankruptcy problem (Aumann and Maschler x ≥ ; 2 0 ð21Þ Table 1 The bankruptcy problem in the Talmud. x ≥ : 3 0 ð22Þ D = 100 D = 200 D = 300 In order for there to be x , x , x that satisfies the constraints, 1 2 3 1 2 3 M = + + + the following formulas must hold from Eqs. (16), (17), (18) and 100 33 1/3 33 1/3 33 1/3 M = 200 50 75 75 the individual rationality (xi ≥ v(i)). M = 300 50 100 150 ÀT ≤ T þ 16 () T ≥ À 8; ð23Þ If a person goes bankrupt leaving a debt of D1, D2 and D3 to each of the creditors 1, 2 and 3, how should the bankrupt’s entire estate (M)be‘fairly’ distributed to each creditor? This table shows ÀT ≤ T þ 10 () T ≥ À 5: ð24Þ an answer to the allocations.

4 HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w ARTICLE

1985). That was accomplished by Aumann and Maschler’s discussion in relation to game theory (Nakayama 1986; Funaki 1989). Aumann and Maschler went on to read from the following description of other passages in the Talmud Mishnah (Baba Metzia 2a). Two hold a garment; one claims it all, the other claims half. Two hold a garment; one claims it all, the other claims half. Then the one is awarded 3/4, the other 1/4. This statement says that if there are those who demand the whole thing and those who demand half of it, it should be divided into 3/4 and 1/4 respectively. This is neither a proportional nor an equal division, but a ‘residual equal division’. For some reason the latter allows 1/2 to the former, so the remaining 1/2 is to be divided equally. Consider the case of two creditors. The amount that Creditor 1 is willing to admit to Creditor 2 is the greater of (M − D1) and 0, M D fi x = max x so ð À 1Þþ,de ned as + ( , 0). Similarly, the amount M D that Creditor 2 is willing to allow to 1 is ð À 2Þþ. The shares of X X Creditors 1 and 2 ( 1 and 2) are as follows fM ÀðM À D Þ ÀðM À D Þ g X ¼ðM À D Þ þ 1 þ 2 þ ; ð31Þ 1 2 þ 2 fM ÀðM À D Þ ÀðM À D Þ g X ¼ðM À D Þ þ 1 þ 2 þ : ð32Þ 2 1 þ 2 They called ‘the Contested Garment principle’ the distribution principle in a bankruptcy case with one bankrupt and two creditors. In the Talmudic bankruptcy problem (three creditors) the CG principle holds for any two creditors and is shown to be Fig. 3 An implementation for ‘interlocked adjuster’. Each cylinder filled ‘nucleolus’ (Aumann and Maschler 1985). with green liquid is connecting blue and yellow cylinders. Due to the interlocking adjuster, if a blue liquid level Ei increases (decreases), the Method corresponding yellow liquid level Ci decreases (increases). Physical method for fair distribution. In this section, we mainly describe how to solve a game of characteristic function form with 4. Then, for each i of yellow liquid cylinder B, add the profit ‘gravity’. First, we have two cylinders A (blue) and B (yellow) generated by the coalition i(+v(i)), and then subtract − v(1, 2, filled with a liquid (incompressibility) such as Fig. 4. The adjus- 3)/3 as shown in Fig. 4c. The current liquid level Ci will ters of A and B are assumed to be interlocked: if one moves, express the complaint of coalition i.ThatisCi ≡ v(i) − xi. the other moves the same amount in the opposite direction (see 5. All six complaints will thereby be expressed and the Fig. 3). By moving the adjusters in sequence according to the preparation will be complete. At this point, the valves instructions as in Fig. 4, each liquid level represents the ‘amount are closed and the adjusters of cylinders A and B are ’ E of complaint of the corresponding coalition. The liquid levels 1, interlocked as shown in Fig. 4d. That is, an increase in ’ E E2 and E3 of the blue liquid correspond to the definitions of cylinder A s i results in a corresponding decrease in the complaints C({2, 3}, x), C({1, 3}, x) and C({1, 2}, x), respectively. corresponding liquid level Ci of cylinder B. Finally, when the two cylinders are set up vertically at the same 6. Stand the two-cylinder systems vertically at the same time time to exert gravity, we find that gravity sets the liquid level in so that gravity works (see Fig. 3). motion and that the final equilibrium liquid level will represent 7. Wait until the equilibrium state is attained and then record the ‘fair distribution’ (physical solution). the final liquid level change yi as shown in Fig. 4d. The procedure for the solution with physics is as follows 8. Note that the liquid level change satisfies the following equation: 1. The valve is open so as to allow for the adjustment of the liquid level by moving the adjuster. For a cylinder of yi ≥ vðiÞÀvð1; 2; 3Þ=3: ð33Þ blue liquid A, divide equally the profit v(1, 2, 3) that would be obtained if everyone had cooperated as shown in Fig. 4a. 9. The distribution is determined by the following equation: Here, xi = v(1, 2, 3)/3. xi ¼ yi þ vð1; 2; 3Þ=3: ð34Þ 2. For each i, add the profit of a two-person coalition other than i (+v(j, k)), and then subtract the profit that would be gained if everyone had cooperated (−v(1, 2, 3)) as shown in Fig. 4b. Physical solutions on fair distribution problems 3. For each i, we record the current liquid level and then set it to the origin of the upcoming gravitational change in the The physical solutions of the taxi problem and the bankruptcy problems (three ways) are shown in Fig. 5. We could find solu- liquid level yi. That is, each distribution will be xi = yi + v(1, 2, 3)/3, satisfying the requirements of the conservation law tions to all the problems using physics. Figure 5a shows the initial liquid levels of the taxi problem, y + y + y = 0. The current liquid level Ei expresses the 1 2 3 which can be set by following the instructions stated in ‘Method’. complaint of the coalition (j, k). That is, Ei ≡ C({j, k}, x) = v Although we only show the cylinder system filled with blue liquid, (j, k) − (xj + xk) = xi − v(1, 2, 3) + v(j, k). the other cylinder system filled with yellow liquid is supposed to

HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w 5 ARTICLE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w

simultaneously, other liquid levels will increase due to inter- locking, resulting in the highest liquid level eventually stop when it cannot further drop. In the case of the taxi problem, the maximum complaint was E2 = E3 = −3 and the equilibrium liquid level was −5. Initial fi E = − E = − levels of the cylinder lled with blue liquid ( 1 9, 2 3 and E3 = −3) could reach the equilibrium level of −5 easily; fi C = C = C =− whereas, those lled with yellow liquid ( 1 2 3 7) reached −11, −5 and −5, respectively. Therefore, the displace- y y y + − − ments 1, 2 and 3 were 4, 2 and 2, respectively. As a result, we obtained the following physical solution xi(i = 1, 2 and 3) from the equation xi = yi + 7 (see ‘Method’). x ; 1 ¼ 11 ð35Þ x ; 2 ¼ 5 ð36Þ x : 3 ¼ 5 ð37Þ According to the physical movements, Person 1 would receive $11 and Person 2 and 3 would both receive the allocation of $5, which were the same results from the calculation of the linear programming problem. Figure 5b shows initial liquid levels of the bankruptcy problem (M = 100 cases). Here the characteristic functions are defined as [(estate(M))−(total debts except S)]+. We obtained v(1) = v(2) = v(3) = v(1, 2) = v(1, 3) = v(2, 3) = 0 and v(1, 2, 3) = 100. In this C = C = C = − case, the maximum complaint was 1 2 3 100/3 and the equilibrium liquid level was Ei = −200/3. Owing to the volume conservation of both liquids, the liquid levels of the Fig. 4 Fair distribution method utilising physics in the game of y y cylinders stopped moving. Therefore, the displacements 1, 2 and characteristic function form. a Equally divide the profit of coalition with all y 3 were all 0. As a result, we obtained the following physical (v(1, 2, 3)). That is, x + x + x = v(1, 2, 3). Here, xi is the allocated profit for 1 2 3 solution xi(i = 1, 2 and 3) from the equation xi = yi + 100/3 (see i i fi j k +v j k . b For each , add the pro t of coalition { , }( ( , )) and subtract the ‘Method’). profit of coalition with all (−v(1, 2, 3)). For each i, mark the present position x = ; as the initial position of displacements yi. Here, xi = yi + v(1, 2, 3)/3, which 1 ¼ 100 3 ð38Þ means y + y + y = 0 due to the volume conservation. Each Ei represents 1 2 3 x = ; the amount of complaint about coalition {j, k}. Here, the complaint Ei ≡ v(j, 2 ¼ 100 3 ð39Þ k) − (xj + xk) = xi − v(1, 2, 3) + v(j, k). c In the yellow cylinder system, for x = : each i, add the profit of coalition {i}(+v(i)) and subtract the profitof 3 ¼ 100 3 ð40Þ −v C coalition with all ( (1, 2, 3)/3). Each i represents the amount of According to the physical movements, Creditors 1, 2 and 3 i C ≡ v i − x complaint about coalition { }. Here, the complaint i ( ) i. d Stand would receive the same allocation of 100/3 (equal division), which vertically this equipment for automatic physical moves caused by gravity. were the same values described in the Talmud (see Table 1). y Owing to the interlocking adjuster, if i increases in the blue cylinder, the Figure 5c shows initial liquid levels of the bankruptcy problem −y adjuster moves to the left-hand side ( i) in the corresponding yellow (M = 200 cases). From the definition of the characteristic func- i y cylinder. For each , read the move (displacement) i from the initial tion, we obtained v(1) = v(2) = v(3) = v(1, 2) = v(1, 3) = 0, v(2, y fi yi ≥ v i − v position. Note that each i must be satis ed ( ( ) (1, 2, 3)/3). Finally, 3) = 100 and v(1, 2, 3) = 200. In this case, the maximum com- we can obtain the distribution xi = yi + v(1, 2, 3)/3. E = − plaint was 1 100/3 and the equilibrium liquid level was Ei = −100. Although the maximum complaint E1 gradually decreased, heading toward the equilibrium liquid level of −100, it ‘ ’ − C − be connected through an interlocking adjuster (if one move, the stopped at 50 because 1 reached 50 from the initial level − E C other moves the same amount in the opposite direction). Each of 200/3 due to the interlocking between 1 and 1. Therefore, E = C x y y y − liquid level represents each complaint, that is, 1 ({2, 3}, ), the displacements 1, 2 and 3 were 50/3, 25/3 and 25/3, E = C x E = C x C = C x C = C x 2 ({1, 3}, ), 3 ({1, 2}, ), 1 ({1}, ), 2 ({2}, ) respectively. As a result, we obtained the following physical and C3 = C({3}, x), respectively. Then, we set up the cylinder solution xi(i = 1, 2 and 3) from the equation xi = yi + 200/3 (see system vertically so that gravity works (Fig. 3). There are various ‘Method’). possibilities depending on whether the ‘interlocking adjusters’ in x ¼ 50; ð41Þ the figure are realised with other liquids or implemented 1 mechanically. In any case, at each level of equilibrium, the tensile x ¼ 75; ð42Þ forces and the pressure inside the liquid (the sum of the statistical 2 and gravitational pressures according to Pascal’s principle) will be x ¼ 75: ð43Þ balanced. 3 In an ideal case, the ‘equal liquid level’ of every cylinder will be According to the physical movements, Creditor 1 would receive realised at equilibrium. However, this system cannot reach the 50 and Creditors 2 and 3 would both receive the allocation of 75, ideal equilibrium state due to the constraints of the adjusters. which were the same values described in the Talmud (see Table 1). The highest liquid level (maximum complaint) will gradually Figure 5d shows initial liquid levels of the bankruptcy pro- decrease, heading towards the equal liquid level. However, blem (M = 300 cases). From the definition of the characteristic

6 HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w ARTICLE

Fig. 5 Physical solutions for profit distribution in a game of characteristic function form. a The taxi problem: Initial liquid levels were E1 = −9, E2 = −3, E3 = −3, C1 = −7, C2 = −7 and C3 = −7, respectively (see ‘Method’). Owing to gravity (stand it vertically), the system could reach equilibrium state E1 = E2 = E3 = −5. Therefore, the displacements from each initial liquid level were y1 =+4, y2 = −2 and y3 = −2, respectively. We obtained the allocations x1 = 11, x2 = 5 and x3 = 5 from xi = yi + 7. Person 1 would receive $11 and Persons 2 and 3 would both receive the allocation of $5 in this case. b The bankruptcy problem (M = 100 cases): initial liquid levels were E1 = E2 = E3 = − 200/3 and C1 = C2 = C3 = −100/3, respectively (see ‘Method’). Owing to the volume conservation of both liquids, the liquid levels of the cylinders stopped moving. Therefore, the displacements y1, y2 and y3 were all 0. We obtained the allocations x1 = x2 = x3 = 100/3 from xi = yi + 100/3. Every creditor would receive the same allocation of 100/3 (equal division). c The bankruptcy problem (M = 200 cases): initial liquid levels were E1 = −100/3, E2 = E3 = − 100 −100/3 and C1 = C2 = C3 = −200/3, respectively (see ‘Method’). Although the maximum complaint E1 gradually decreased, heading toward the equilibrium level of −100, it stopped at the level of −50 because C1 also reached −50 from the initial liquid level of −200/3 due to the interlocking between E1 and C1. Therefore, the displacements y1, y2 and y3 were −50/3, 25/3 and 25/3, respectively. We obtained the allocations x1 = 50, x2 = 75 and x3 = 75 from xi = yi + 200/3. Creditor 1 would receive 50 and Creditors 2 and 3 would both receive the allocation of 75. d The bankruptcy problem (M = 300 cases): Initial liquid levels were E1 = 0, E2 = 100, E3 = 200 and C1 = C2 = C3 = − 100, respectively (see ‘Method’). Although the maximum complaint E1 gradually decreased, heading toward the equilibrium level of −100, it stopped at the level of −50 because C1 also reached −50 from the initial level of −100 due to the interlocking between E1 and C1. Therefore, the displacements y1, y2 and y3 were −50, 0 and +50, respectively. As a result, we obtained the allocations x1 = 50, x2 = 100 and x3 = 150 from the equation xi = yi + 100. Creditors 1, 2 and 3 would receive 50, 100 and 150, respectively (proportional division). function, we obtained v(1) = v(2) = v(3) = v(1, 2) = 0, v(1, 100 (see ‘Method’). = v = v = 3) 100, (2, 3) 200 and (1,2,3) 300. In this case, the x ; E = 1 ¼ 50 ð44Þ maximum complaint was 1 0 and the equilibrium liquid level was Ei = −100. Although the maximum complaint E 1 x ¼ ; ð Þ gradually decreased, heading toward the equilibrium liquid 2 100 45 level of −100, it stopped at −50 because C reached −50 from 1 x ¼ : ð Þ the initial liquid level of −100 due to the interlocking between 3 150 46 E C y y y − 1 and 1. Therefore, the displacements 1, 2 and 3 were 50, According to the physical movements, the creditors would 0and+50, respectively. As a result, we obtained the following receive 50, 100 and 150, respectively (proportional division), physical solution xi(i = 1, 2 and 3) from the equation xi = yi + which were the same values described in the Talmud (see Table 1).

HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w 7 ARTICLE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w

Using four practical and simple examples, with N = 3 persons, Humans are yet to come up with a computational alternative we showed that a fair distribution in games of characteristic to the digital computer. Recently computational methods that function form can be physically obtained using the proposed exploit physical characteristics, such as quantum computing, method. We can also obtain the physical solutions in general have been proposed, but they have not been generalised or N-person problems provided we successfully implement the theorised. interlocking adjusters. In general N-person problems, the number Kim et al. have been trying to extract computational methods of variables (complaints) increases exponentially (2N − 2), that is, that exploit physical characteristics, such as fluctuations (ran- 6, 14, 30, 62, 126, … , and the physical correspondences become domness), conservation laws (correlation) and analogue values complicated and difficult to understand. Therefore, more com- (cardinality of the continuum), not only in quantum physics but plex examples will be dealt with in our future work. also in the framework of classical physics (Kim et al. 2016, 2015, 2013;Naruseetal.2015;Kimetal.2016). They have also developed devices that can be implemented using various Conclusions physical systems, and in which the physics itself makes its own In this study, we have shown that the fair distribution of profitin computational decisions (Naruse et al. 2017;Tsuchiyaetal. games of characteristic function form can be solved utilising 2018;Kimetal.2019;Maetal.2019). physics. More specifically, the linear programming problem used This paper illustrates based on four simple examples that to compute ‘nucleolus’ can be regarded as a physical system that the calculation of fair distribution can be realised directly by responds to gravity, and the solution can be found efficiently. exploiting physics. Extending beyond N = 4isachallenge Although it is only shown to be effective for solving relatively for future works, but if we can generalise it, we can implement simpler problems with N = 3, we can deduce that if we could it not only in the liquid systems but also in many other physically construct the system for general N-person problems, physical systems, which will impact humanity. For example, if we could always obtain a physical solution of the fair distribution. it is implemented on a semiconductor chip (using analogue We plan to consider more complex examples in future. Identifi- resistance values (Kim et al. 2019)), it could be linked to GPS cation of the necessary conditions, such as physical parameters and big data, so that the distance from your current location and constraints, for this physical solution to work, is also to your home is immediately known, helping us decide in an future works. instant how much to spend on a taxi in case of riding In order for this approach to become a candidate, a viable with friends (taxi problem), enabling us to make our own method, for overcoming the computational difficulties in digital decisions. computers, it is necessary to at least address the general N-person The distribution of the bankruptcy problem can also be problems, and how the solution should be presented. Never- calculated immediately, which should reduce disputes and theless, the employment of such a physical solution may have the conflicts based on inaccurate information and misunderstand- following implications and prospects: ings, eliminating fraud and injustice. Further, decisions can be made on the basis of ‘information based on accurate calcula- 1. Able to achieve computational reduction. tions’, and the resources of thought that were previously taken 2. Able to make real-time decisions at the time of information up by the problem itself can now be used for higher-order updates, e.g., parameter changes during calculations fl problems (e.g., meta-thinking, which considers even the reac- ( exibility). tion of the other party after the decision has been made for self- 3. Able to interact with information between different species. interests next time). 4. Able to interlock the distribution motion directly from Natural intelligence (NI) (Kim et al. 2016, 2017)meansthat physical calculations (water distribution, oil distribution) and fi intelligent functions emerge from computations directly using 5. Eliminate arti ciality by using natural phenomena, thus the underlying physics and physical characteristics, such as making it easier for people to understand and to reach fluctuations (randomness), conservation laws (correlation) and consensus (compromise). analogue values (cardinality of the continuum). The proposed Real-world problems, such as the distance between two people method based on NI provides solutions and physical imple- in a taxi problem, must be addressed often without complete mentations, as well as a completely new perspective to social information available. We should be flexible and able to respond sciences, which is a totally different way of AI (digital com- flexibly if the information is updated along the way. This system puting). To date, the winners of the excessive competition have will become a candidate for overcoming the computational dif- reaped short-term benefits, but in the end, thanks to their ficulties in such problems and will provide a new perspective to disdain for the overall benefit, the bill has returned to them. As open up a physics-based policymaker that is adaptable to real- the social implementation of NI progresses, we can see the time changes. negative consequences of our overly selfish behaviour by referring to the results of instantaneous physical computations when deciding our actions, and we will be able to change our Discussion and future developments strategy to one of cooperation rather than competition. The role of computation in our society is increasing daily. Despite Importantly, a change in strategic direction—co-operation the importance of computation, there are serious limitations to rather than competition—will also come from self-interest the existing computational methods, including AI. For instance, (long-term) strategies because one can realise owing to NI that consider the 19 × 19 board game of Go. The number of self- for self-interest in the long term, one cannot avoid taking into avoiding routes from one end to the bottom right diagonal end is account the collective interests. Collective interests will be greater than the number of atoms in the universe (1087 >1080). guaranteed as a system, enabling humanity to live in a higher Surprisingly, this number could not be calculated until recently, order of thinking. It will be very interesting to see what new when a new theory called ‘the theory of counting’ was developed social problems will arise in such a case. (Iwashita et al. 2012). Even the things that lie in the ordinary course of our daily lives are full of things that we cannot calculate. When it comes to our ability to computing, we are almost Data availability powerless, especially when considering solving societal problems. We generated and analysed no data in this study.

8 HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-021-00806-w ARTICLE

Received: 19 February 2021; Accepted: 11 May 2021; Pastreich E (2020) I shall fear no evil: declaring the candidacy for the U.S. pre- sidential election 2020. In: Kim S-J (trans). Design Egg, Tokyo Piketty T (2014) Capital in the twenty-first century (trans: Goldhammer A). Harvard University Press, Cambridge Saeki Y (1980) The logic of decision: an invitation to social decision theory. University of Tokyo Press, Tokyo References Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl – Arrow KJ (1951) Social choice and individual values. Yale University Press, New Math 17(6):1163 1170 Haven Steinhaus H (1960) Mathematical snapshot. Oxford University Press, Oxford Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem Tsuchiya T et al. (2018) Ionic decision-maker created as novel, solid-state devices. from the Talmud. J Economic Theory 36(2):195–213 Sci Adv 4(9):eaau2057 fi fi Brandt F et al. (eds) (2016) Handbook of computational social choice. Cambridge Yamamoto H, Suzuki T, Umetani R (2020) Justi ed defection is neither justi ed fi University Press, New York nor unjusti ed in indirect reciprocity. PLoS ONE 15(6):e0235137 – Endriss U (ed.) (2017) Trends in computational social choice. AI Access Yokoo M et al. (2013) The cooperation game. Comput Softw 30:33 51 Funaki Y (1989) The nucleolus and Sharpeley value of cooperative games: on the Zheng S et al. (2020) The AI economist: improving equality and productivity with bankruptcy issue. Oper Res 34(11):582–587 AI-driven tax policies. arXiv:2004.13332 Gilles RP (2010) The cooperative game theory of networks and hierarchies. Springer-Verlag, Berlin Heidelberg Acknowledgements Harari YN (2020) In the battle against coronavirus, humanity lacks leadership. This work was supported by the research grant SP001 from SOBIN Institute, LLC. We Time (March 15th). https://time.com/5803225/yuval-noah-harari-coronavirus- would like to thank Dr. Emanuel Pastreich (President of the Asia Institute) and Dr. humanity-leadership/ Alexander Krabbe (the Asia Institute, Germany) for fruitful discussions on this work at Iwashita H, Kawahara J, Minato S (2012) ZDD-based computation of the number of an early stage. paths in a graph. Hokkaido University, TCS Technical Reports: TCS-TR-A-12-60 Jinno N (2010) The Economics of OMSORG. Iwanami Shoten, Tokyo Karmarkar N (1984) A new polynomial time algorithm for linear programming. Author contributions Combinatorica 4:373–395 S.-J.K. invented the method. All authors analysed the results. S.-J.K. wrote the manuscript Kim S-J et al. (2013) Decision maker based on nanoscale photo-excitation transfer. in discussion with T.T. and K.S. Sci Rep 3:2370 Kim S-J et al. (2016) Decision maker based on atomic switches. AIMS Mater Sci 3(1):245–259 Competing interests Kim S-J, Aono M, Nameda E (2015) Efficient decision-making by volume- The authors declare no competing interests. conserving physical objects. New J Phys 17:083023 Kim S-J, Naruse M, Aono M (2016) Harnessing the computational power of fluids Additional information – for the optimization of collective decision making. Philosophies 1(3):245 260 Correspondence and requests for materials should be addressed to S.-J.K. Kim S-J, Naruse M, Aono M (2017) From artificial intelligence to natural intelli- gence: exploitation of hidden computational ability in nature. Trans IEICE Reprints and permission information is available at http://www.nature.com/reprints J100-C:261–268 Kim S-J et al. (2019) Reinforcement learning system comprising resistive analog Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in neuromorphic devices. IEEE Xplore. https://doi.org/10.1109/IRPS.2019.8720428 published maps and institutional affiliations. Kishimoto S (2015) Introduction to cooperative game theory. Oper Res 60(6):343–350 Lempert R (2002) Agent-based modeling as organizational and public policy simulators. PNAS 99(3):7195–7196 Levi M (2009) The mathematical mechanic: using physical reasoning to solve problems. Princeton University Press, New Jersey Open Access This article is licensed under a Creative Commons Ma J et al. (2019) A reinforcement-learning-based distributed resource selection Attribution 4.0 International License, which permits use, sharing, algorithm for massive IoT. Appl Sci 9(18):3730 adaptation, distribution and reproduction in any medium or format, as long as you give Nakayama M (1986) The bankruptcy problem and the nucleolus of the n-game: the appropriate credit to the original author(s) and the source, provide a link to the Creative allocation principle of Talmud. J Econ Stud 32:315–328 Commons license, and indicate if changes were made. The images or other third party ’ Naruse M et al. (2015) Single-photon decision maker. Sci Rep 5:13253 material in this article are included in the article s Creative Commons license, unless Naruse M et al. (2017) Ultrafast photonic reinforcement learning based on laser indicated otherwise in a credit line to the material. If material is not included in the chaos. Sci Rep 7:8772 article’s Creative Commons license and your intended use is not permitted by statutory Neumann J, Morgenstern O (2007) Theory of games and economic behavior (60th regulation or exceeds the permitted use, you will need to obtain permission directly from Anniversary Commemorative Edition). Princeton University Press, New Jersey the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Nowak MA (2006) Five rules for the evolution of cooperation. Science 314 licenses/by/4.0/. (5805):1560–1563 Pastreich E (2019) Farewell to arms: the crisis of global warming and the ninth article of the constitution. In: Kawase S (ed.). Toho Publishing, Tokyo © The Author(s) 2021, corrected publication 2021

HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | (2021) 8:131 | https://doi.org/10.1057/s41599-021-00806-w 9